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Logic

This presentation introduces the basics of first-order predicate calculus, including logical statements, axioms, and inference rules. Examples and explanations are provided to aid understanding.

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Logic

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  1. Logic We begin with some preliminaries on mathematical logic: first-order predicate calculus (a formal way of expressing logical statements). 01/21/98 Reference: Programming Languages by Louden Logic Slide -1

  2. #1: “0 is a natural number” becomes natural (0). #2: “2 is a natural number” becomes natural (2). #3: “-1 is a natural number”becomes natural (-1). (false) #4: “for all x, if x is a natural number then so is the successor of x.” becomes "x, natural (x) natural (successor (x)). #1 and #4 are assumed true; they are axioms. #2 can be deduced from the axioms since 2 is the successor of the successor of 0. Assertion #3 cannot be proved from the axioms and is assumed to be false. 01/21/98 Reference: Programming Languages by Louden Logic Slide -2

  3. First-Order Predicate Calculus uses the following concepts: • Constants :numbers or names, also called atoms. • Predicates :functions that return true or false. • Functions :other functions • Variables :quantities that are not yet given a specific value 01/21/98 Reference: Programming Languages by Louden Logic Slide -3

  4. Connectives : •  (and), •  (or), •  (implies, also denoted ), •  (is implied by, also denoted ), •  (if and only if, also denoted ), •  (not) • Quantifiers : (for all),  (there exists) • Punctuation : comma, period, parentheses. 01/21/98 Reference: Programming Languages by Louden Logic Slide -4

  5. Now we can write: mammal (horse). mammal (human). x, mammal (x ) (legs (x, 4) arms (x, 0))  (legs (x, 2) arms (x, 2)). arms (horse, 0). arms (human, 0). arms (human, 2). arms(horse, 2). 01/21/98 Reference: Programming Languages by Louden Logic Slide -5

  6. We can deduce legs (horse, 4). From x, mammal (x ) (legs (x, 4) arms (x, 0))  (legs (x, 2) arms (x, 2)). And mammal (horse). And arms (horse, 0). And arms(horse, 2). • We need a systematic notation which allows us to develop automatic computerized proof system. • One successful approach is based on Horn clauses. 01/21/98 Reference: Programming Languages by Louden Logic Slide -6

  7. Alfred Horn studied a restricted form of inference rules. • Inference rules are ways of deriving or proving new statements from a list of statements. • It is not as expressive as full predicate calculus but it does handle a large number of logical statements. • Horn clauses have the form a1 a2  ... an b 01/21/98 Reference: Programming Languages by Louden Logic Slide -7

  8. a1 a2  ... an b • The left hand side consists of a conjunction (sequence of “and’s”) and there is only one conclusion. • Axioms or facts are written  b • The opposite of a fact is a query. b  01/21/98 Reference: Programming Languages by Louden Logic Slide -8

  9. When there are variables in a Horn clause that appear on both sides of the we assume that there is a “for all” for every variable at the front of the clause. • Hence (u > 0)  (v < 0) (v < u). means u, v, (u > 0)  (v < 0)  (v < u). 01/21/98 Reference: Programming Languages by Louden Logic Slide -9

  10. If there are variables that appear on the left of the  only, then we assume there is a “there exists” for all such variables. (u > z)  (z > v)  (u > v). means u, v, z (u > z)  (z > v)  (u > v). 01/21/98 Reference: Programming Languages by Louden Logic Slide -10

  11. The following example in Prolog, grandparent( X , Y ) :– parent(X , Z),parent(Z,Y). , could be interpreted as, “x is a grandparent of y if x is the parent of someone who is a parent of y.” This Prolog example translates to u, v, z , parent(u, z )  parent(z, v ) grandparent(u, v ). Rewritten as a Horn clause it has the following form: parent(u, z )  parent(z, v ) grandparent(u, v ). 01/21/98 Reference: Programming Languages by Louden Logic Slide -11

  12. In order to start an automated reasoning process, Horn clauses are written in reverse: (u > v)  (u > z )  (z > v ). grandparent(u, v )  parent(u, z )  parent(z, v ). • A query or goal is written  grandparent( x, y ). • We can have a conjunction of several queries, which form a compound query: parent(x, z ), parent(z, y ). 01/21/98 Reference: Programming Languages by Louden Logic Slide -12

  13. Now the semantic effect of procedures can be specified as one or more Horn clauses: sort(x, y)  permutation(x, y)  sorted(y). • This line specifies “sort.” “permutation” and “sorted” are a bit difficult to express but we will see how to do it in Prolog. • Next we drop the symbol  and write simply a comma. • A fact in this format is sort(empty-list, empty-list )  . 01/21/98 Reference: Programming Languages by Louden Logic Slide -13

  14. Resolution and Unification • Resolution is an efficient inference rule, attributed to Alan Robinson, now retired from Syracuse University. Given a  a1 , a2 , ... , an b  b1 , b2 , ... , bm • Unification is the process of pattern matching to make statements identical. • When a variable is set equal to a pattern the variable is said to be instantiated. 01/21/98 Reference: Programming Languages by Louden Logic Slide -14

  15. If bimatches a then we can infer b b1 , b2, ... , bi - 1 , a1 , a2 , ... , an, bi + 1 , ... , bm • If we have a goal  a and if we have a Horn clause a  a1 , a2 , ... , an then resolution replaces the original goal by  a1 , a2 , ... , an a sequence of subgoals. 01/21/98 Reference: Programming Languages by Louden Logic Slide -15

  16. Simplest case is when the goal is already a known fact: mammal(horse). Query: • mammal(horse). Using resolution, the two Horn clauses are combined mammal(horse)  mammal(horse). Canceling both sides we get: . This is regarded as a success.

  17. More complicated example: • Horn clauses: legs(x, 2)  mammal(x), arms(x, 2). legs(x, 4)  mammal(x), arms(x, 0). mammal(horse) . arms(horse, 0) . • Query: legs(horse, 4). 01/21/98 Reference: Programming Languages by Louden Logic Slide -16

  18. Applying resolution requires a match x = horse in the second Horn clause then we can substitute the query by the subgoals:  mammal(horse), arms(horse, 0). • We can resolve between this subgoal and the clause mammal(horse)  . • giving  arms(horse, 0). 01/21/98 Reference: Programming Languages by Louden Logic Slide -17

  19. Resolving with the last input Horn clause:  arms(horse, 0). arms(horse, 0) . gives  which is regarded as success. 01/21/98 Reference: Programming Languages by Louden Logic Slide -18

  20. A little extra background may help to see exactly why goals are written “a.” and why reaching “.” is regarded as success. • First recall the truth table for “is implied by:” P Q P  Q T T T T F T F T F F F T 01/21/98 Reference: Programming Languages by Louden Logic Slide -19

  21. The only case where “is implied by” is false, is when we are asserting that a false statement is implied by a true statement. • Next, we shall write facts and goals differently. A fact is always true and is therefore “implied by” T, since T cannot imply F: a T. 01/21/98 Reference: Programming Languages by Louden Logic Slide -20

  22. The next issue is that if we wanted to deduce that b is true, then we could equally show that F b. is false, since F b. is false exactly when b is true. • The inference technique using resolution and inference is to include the Horn clause F b. with the others in the program ... • and then see if there is a contradiction. • If adding this clause causes a contradiction, then b must have been true. 01/21/98 Reference: Programming Languages by Louden Logic Slide -21

  23. How do we detect a contradition? We simply add F b. and then try to deduce that F  T. which is known to be false. • This is why “.” is regarded as success. • It really means that we have arrived at F  T. If we add F  legs(horse, 4). to the set of Horn clauses. F  legs(horse, 4). must have been false, i.e. legs(horse, 4) must be true. 01/21/98 Reference: Programming Languages by Louden Logic Slide -22

  24. For example, given 1) legs(x, 2)  mammal(x), arms(x, 2). 2) legs(x, 4)  mammal(x), arms(x, 0). 3) mammal(horse) T. 4) arms(horse, 0) T. 5) F legs(horse, 4). • resolve between 2) and 5) unifying x = horse to obtain 6) F  mammal(horse), arms(horse, 0). • resolve between 3) and 6) to obtain 7) F T, arms(horse, 0). 01/21/98 Reference: Programming Languages by Louden Logic Slide -23

  25. 4) arms(horse, 0) T. 7) F T, arms(horse, 0). • resolve between 4) and 7) to obtain: 8) F  T, T. i.e. the contradiction F  T. and T, T normally written T  T is true. 01/21/98 Reference: Programming Languages by Louden Logic Slide -24

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